A NI-Republic border ought to happen, as Northern Ireland will be outside the EU following Brexit. There’s no basis in law that I am aware of to suppose that Northern Ireland would get special EU status, although I understand that the EU is willing to consider it – thereby giving more thought to the province than Westminster has since the EU referendum was mooted.
Technically, Brexit breaks the Good Friday Agreement, under which the border is invisible. In addition, during the Brexit vote in Parliament, an amendment to Article 50 which would have protected the Good Friday Agreement during the Brexit process was comprehensively voted down – effectively, Parliament voted to break GFA.
Complicating the issue even more is the historic Common Travel Area between the UK and Ireland. This works just fine if both the UK and Ireland are in the EU, or outside the EU. No one knows how this will operate, or even if it can, after Brexit.
Then, there’s Gibraltar, where similar issues apply.
~Sigh~. Look before you leap. Actions have consequences. If it ain’t broke, don’t fix it. Too many spanners, not enough broth.